The generalized distance spectrum of a graph and applications

ABSTRACT

The generalized distance matrix of a graph is the matrix whose entries depend only on the pairwise distances between vertices, and the generalized distance spectrum is the set of eigenvalues of this matrix. This framework generalizes many of the commonly studied spectra of graphs. We show that for a large class of graphs these eigenvalues can be computed explicitly. We also present the applications of our results to competition models in ecology and rapidly mixing Markov chains.

KEYWORDS:

• Distance-regular graph
• spectral graph theory
• ecological models
• Markov chains

2010 MATHEMATICS SUBJECT CLASSIFICATIONS:

• 05C50
• 92D25
• 60J10
• 60J22
• 60J27

Disclosure statement

No potential conflict of interest was reported by the author(s).

Notes

1 It is common in this context to use λ to denote the eigenvalues of P, but we have another use for λ below.